## Long-Term Trend Student Sampling

Within selected schools, the school was asked to list all students with the appropriate birth dates for the specific age for which the school was selected. If a school selected for the Age 9 or Age 13 sample had 128 or less students, all age-eligible students were selected into the sample for that school. Otherwise, a sample of 128 age-eligible students was selected. For schools selected for the Age 17 sample, the cutoff was 121 rather than 128.

As discussed in Targeted Students by Age, there were four different sessions to which sampled students could be assigned. The relative targeted number of students for each session are shown in the tables below, which present the targets as percentages, and as fractions of the maximal within-school sample sizes (128 for the Age 9 and Age 13 samples, and 121 for the Age 17 sample), rounded to the nearest integer.

Session-level targets for within-school sample, long-term trend study: By session type, 2004
Sessions Target percentage of students: Age 9
and 13
Maximal sample size: Age 9 and 13 Target percentage of students: Age 17 Maximal sample size: Age 17
Total 100.0 128 100.0 121
Session B: Operational and pilot sessions 62.4 80 67.0 81
Sessions C and D: Math bridge sessions 20.5 26 14.7 18
Session F: Reading Bridge session 17.1 22 18.4 22
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2004 Long-Term Trend Assessment.

In the Age 9 and Age 13 studies, schools with the maximal student sample size of 128 were randomly allocated to either receive a ‘C’ session or a ‘D’ session, with one-half probability on each option. The school received a ‘B’ session and ‘F’ session with certainty. If the school was selected for a ‘C’ session, a random integer in the closed interval [1,64] was selected with equal probability. This random integer determined the starting point on a fixed sequence of 40 B’s,13 C’s, and 11 F’s, called the ’64-string’. Sampled students were assigned to sessions based on this selected sequence. This approach effectively assigns 80 students to ‘B’, 26 to ‘C’, and 22 to ‘F’ as desired. If the school was selected for a ‘D’ session, a similar approach resulted in a fixed sequence of 40 B’s, 13 D’s, and 11 F’s (the same fixed sequence as the C’s, except that the D’s replace the C’s).

In schools with less than 128 students, the sample design was set to achieve the same percentages, but was also set to avoid a too-few number of students being assigned to any session. The general minimum per session was set at 15 (unless there were less than 45 students, in which case the minimum was 12). (Note that it was allowable to have zero students for a session, but if there were any students in the session, the total was 15 or greater.) In all cases, the school was randomly selected to be a ‘C’ session or a ‘D’ session math bridge school, with ½ probability on each session. The 64-string for a ‘C’ school or the 64-string for a ‘D’ was then utilized as follows:

• 88 or more students: a starting point (1 to 64) was selected on the 64-string, and students were assigned using the 64-string from the random starting point (winding around back to the starting point once). Every session ('B'; 'C' or 'D'; 'F') had at least one student.

• 75 to 87 students: a starting point was randomly selected on the 64-string, and a 5-string was then drawn starting from the random starting point. This 5-string was replicated as many times as was needed, and students were assigned according to the replicated 5-string. Note that this guaranteed any session represented on the 5-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ or ‘D’ session with probability 61/64 (with ½ chance of ‘C’ and ½ chance of ‘D’), and the ‘F’ session with probability 55/64.

• 60 to 74 students: a starting point was randomly selected on the 64-string, and a 4-string starting from the random starting point was drawn. This 4-string was replicated as many times as needed, and students were assigned according to the replicated 4-string. Note that this guaranteed any session represented on the 4-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ or ‘D’ session with probability 50/64 (with ½ chance of ‘C’ and ½ chance of ‘D’), and the ‘F’ session with probability 44/64.

• 45 to 59 students: a starting point was randomly selected on the 64-string, and a 3-string starting from the random starting point was drawn. This 3-string was replicated as many times as needed, and students were assigned according to the replicated 3-string. Note that this guaranteed any session represented on the 3-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ or ‘D’ session with probability 39/64 (with ½ chance of ‘C’ and ½ chance of ‘D’), and the ‘F’ session with probability 33/64.

• 24 to 44 students: a starting point was randomly selected on the 64-string, and a 2-string starting from the random starting point was drawn. This 2-string was replicated as many times as needed, and students were assigned according to the replicated 2-string. Note this guaranteed any session represented on the 2-string will have at least 12 students. The ‘B’ session was included with probability 1, the ‘C’ or ‘D’ session with probability 26/64 (with ½ chance of ‘C’ and ½ chance of ‘D’), and the ‘F’ session with probability 22/64.

• Less than 24 students: a starting point was randomly selected on the 64-string, and a 1-string starting from the random starting point was drawn. This is the session that was assigned to every student. The ‘B’ session was this session with probability 40/64, the ‘C’ or ‘D’ session with probability 13/64 (with ½ chance of ‘C’ and ½ chance of ‘D’), and the ‘F’ session with probability 11/64.

In the Age 17 study, schools with the maximal student sample size of 121 were randomly allocated to receive a ‘B’ session, a 'C' session, and an ‘F’ session with certainty. A random integer from 1 to 61 was drawn to determine a starting point on a fixed sequence of 41 B’s, 9 C’s, and 11 F’s, called the ’61-string’. The final expected percentages were 41/61 students assigned the 'B' session, 9/61 students assigned the 'C' session, and 11/61 students assigned the 'F' session.

In schools with less than 121 students, the sample design was set to achieve the same percentages, but was also set to avoid a too-few number of students being assigned to any session. The general minimum per session was set at 15 (unless there were less than 45 students, in which case the minimum was 12). (Note that it was allowable to have zero students for a session, but if there were any students in the session, the total was 15 or greater.) The 61-string was utilized as follows:

• 105 or more students: a starting point was randomly selected on the 61-string, and students were assigned using the sequence from the random starting point (winding around back to the starting point once). All three sessions ('B', 'C', 'F') had at least one student.

• 75 to 104 students: a starting point was randomly on the 61-string, and a 5-string starting from the random starting point was drawn. This 5-string was replicated as many times as needed, and students were assigned according to this replicated 5-string. Note that this guaranteed that any session represented on the 5-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ session with probability 45/61, and the ‘F’ session with probability 51/61.

• 60 to 74 students: a starting point was randomly on the 61-string, and a 4-string starting from the random starting point was drawn. This 4-string was replicated as many times as needed, and students were assigned according to this replicated 4-string. Note that this guaranteed that any session represented on the 4-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ session with probability 36/61, and the ‘F’ session with probability 42/61.

• 45 to 59 students: a starting point was randomly on the 61-string, and a 3-string starting from the random starting point was drawn. This 3-string was replicated as many times as needed, and students were assigned according to this replicated 3-string. Note that this guaranteed that any session represented on the 3-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ session with probability 27/61, and the ‘F’ session with probability 33/61.

• 24 to 44 students: a starting point was randomly on the 61-string, and a 2-string starting from the random starting point was drawn. This 2-string was replicated as many times as needed, and students were assigned according to this replicated 2-string. Note that this guaranteed that any session represented on the 2-string had at least 15 students. The ‘B’ session was included with probability 1, the ‘C’ session with probability 18/61, and the ‘F’ session with probability 22/61.

• Less than 24 students: a starting point was randomly on the 61-string, and a 1-string starting from the random starting point was drawn. This session should be assigned to every student. The ‘B’ session was included with probability 41/61, the ‘C’ session with probability 9/61, and the ‘F’ session with probability 11/61.

Last updated 16 March 2009 (FW)